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Search: id:A033954
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| A033954 |
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n*(4*n+3). Also, second 10-gonal (or decagonal) numbers. |
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+0
18
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| 0, 7, 22, 45, 76, 115, 162, 217, 280, 351, 430, 517, 612, 715, 826, 945, 1072, 1207, 1350, 1501, 1660, 1827, 2002, 2185, 2376, 2575, 2782, 2997, 3220, 3451, 3690, 3937, 4192, 4455, 4726, 5005, 5292, 5587, 5890, 6201, 6520, 6847, 7182, 7525, 7876, 8235
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Write 0,1,2,... in clockwise spiral; sequence gives numbers on positive x axis.
The equations 1 + 2 = 3 and 3^2 + 4^2 = 5^2 set the stage for considering whether it is also true that 5^3 + 6^3 = 7^3 and 7^4 + 8^4 = 9^4. Reflecting on Fermat's Last Theorem or resorting to a calculator dispels any hope that either of the two equations could be correct. However, it is true that 5^3 + 6^3 + 2 = 7^3 and 7^4 + 8^4 + 64 = 9^4. More significantly, each of these equations is the first of an infinite sequence of equations featuring consecutive integers that conform to the spirit of the equations mentioned in A000384. For n>0, a(n)^4+(a(n)+1)^4 +...+(a(n)+n)^4 +(4*A000217(n))^3 = (a(n)+n+1)^4+...+(a(n)+2n)^4; e.g., 7^4+8^4+(4*1)^3=9^4; 22^4+23^4+24^4+(4*3)^3=25^4+26^4; see also 045944 - Charlie Marion (charliemath(AT)optonline.net), Dec 8 2007
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REFERENCES
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S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 1998) 188; 30 (#4, 1999-2000), 246-250.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
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LINKS
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Emilio Apricena, A version of the Ulam spiral
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FORMULA
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G.f.: x(7+x)/(1-x)^3. - Michael Somos, Mar 03 2003
a(n)=8*n+a(n-1)-9 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 12 2009]
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EXAMPLE
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16 17 18 19 ...
15 4 5 6 ...
14 3 0 7 ...
13 2 1 8 ...
For n=2, a(2)=8*2+0-9=7; n=3, a(3=8*3+7-9=22; n=4, a(4)=8*4+22-9=45 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 12 2009]
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MATHEMATICA
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s=0; lst={s}; Do[s+=n++ +7; AppendTo[lst, s], {n, 0, 7!, 8}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 16 2008]
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PROGRAM
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(PARI) a(n)=4*n^2+3*n
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CROSSREFS
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Same as A033951 except start at 0. Cf. A002943.
a(n)=A001107(-n)=A074377(2n).
Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.
Cf. A002620.
Sequence in context: A047718 A031053 A063130 this_sequence A159227 A081274 A038764
Adjacent sequences: A033951 A033952 A033953 this_sequence A033955 A033956 A033957
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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