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Search: id:A002412
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| A002412 |
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Hexagonal pyramidal numbers, or greengrocer's numbers.
(Formerly M4374 N1839)
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+0
22
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| 0, 1, 7, 22, 50, 95, 161, 252, 372, 525, 715, 946, 1222, 1547, 1925, 2360, 2856, 3417, 4047, 4750, 5530, 6391, 7337, 8372, 9500, 10725, 12051, 13482, 15022, 16675, 18445, 20336, 22352, 24497, 26775, 29190, 31746, 34447, 37297, 40300
(list; graph; listen)
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OFFSET
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0,3
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REFERENCES
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A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 2.
T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.
I. Siap, Linear codes over F_2 + u*F_2 and their complete weight enumerators, in Codes and Designs (Ohio State, May 18, 2000), pp. 259-271. De Gruyter, 2002.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
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a(n) = n*(n+1)*(4*n-1)/6. G.f.: x*(1+3*x)/(1-x)^4.
n^3-sum(i^2, i=1..(n-1)) - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de).
Partial sums of n odd triangular numbers, e.g. a(3)=t(1)+t(3)+t(5)=1+6+15=22 - Jon Perry (perry(AT)globalnet.co.uk), Jul 23 2003
a(n)=sum(i=0, n-1, (n-i)(n+i)) - Jon Perry (perry(AT)globalnet.co.uk), Sep 26 2004
Binomial transform of (1, 6, 9, 4, 0, 0, 0,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 16 2007
a(n) = n*A000292(n) - (n-1)*A000292(n-1) = n*C((n+2),3) - (n-1)*C((n+1),3); e.g. a(5) = 95 = 5*35 - 4*20. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 28 2007
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MAPLE
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seq(add((n^2-k^2), k=0..n), n=0..39); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 02 2006
a:=n->sum(binomial(n+j, 1)*binomial(n-j, 1), j=0..n): seq(a(n), n=0..39); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 02 2006
A002412:=z*(1+3*z)/(z-1)**4; [S. Plouffe in his 1992 dissertation.]
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PROGRAM
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(PARI) v=vector(40, i, t(i)); s=0; forstep(i=1, 40, 2, s+=v[i]; print1(s", "))
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CROSSREFS
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Bisection of A002623. Equals A000578(n)-A000330(n-1).
Cf. A016061.
a(n)= A093561(n+2, 3), (4, 1)-Pascal column.
Cf. A000292.
Sequence in context: A010001 A014073 A129109 this_sequence A041215 A060822 A011926
Adjacent sequences: A002409 A002410 A002411 this_sequence A002413 A002414 A002415
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KEYWORD
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nonn,easy,nice
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AUTHOR
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njas
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EXTENSIONS
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Plouffe Maple line edited by R. J. Mathar, May 16 2008
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