A213630 a(t_1 t_2...t_n) = (t_1 + t_2)*t_1 t_2 + ... + (t_n-1 + t_n)*t_n-1 t_n.
1, 4, 9, 16, 25, 36, 49, 64, 81, 10, 22, 36, 52, 70, 90, 112, 136, 162, 190, 40, 63, 88, 115, 144, 175, 208, 243, 280, 319, 90, 124, 160, 198, 238, 280, 324, 370, 418, 468, 160, 205, 252, 301, 352, 405, 460, 517, 576, 637, 250, 306, 364, 424, 486, 550, 616
Offset: 1
Examples
a(123456) = (1+2)*12 + (3+4)*34 + (5+6)*56 = 890. a(890) = (0+8)*08 + (9+0)*90 = 874. a(15) = (1+5)*15 = 90. a(7) = (0+7)*07 = 49.
Links
- Felipe Bottega Diniz and T. D. Noe, Table of n, a(n) for n = 1..1000 (first 100 terms from Felipe Bottega Diniz)
- F. B. Diniz, About a New Family of Sequences, arXiv:1607.06082 [math.GM], 2016.
Programs
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Maple
read("transforms"): A213630 := proc(n) local dgs,a,i ; if n < 10 then n^2; else dgs := convert(n,base,10) ; a := 0 ; for i from 2 to nops(dgs) do a := a+(op(i-1,dgs)+op(i,dgs))*digcat2(op(i,dgs),op(i-1,dgs)) ; end do: a; end if; end proc: # R. J. Mathar, Aug 10 2012 -
Mathematica
num[n_] := Module[{d = IntegerDigits[n], d2}, If[OddQ[Length[d]], d = Prepend[d, 0]]; d2 = Partition[d, 2]; Sum[(d2[[i, 1]] + d2[[i, 2]])*(10*d2[[i, 1]] + d2[[i, 2]]), {i, Length[d2]}]]; Table[num[n], {n, 100}] (* T. D. Noe, Jun 19 2012 *)
Formula
Let t_1t_2...t_n be a natural number.
a(t_1t_2...t_n) = (t_1 + t_2)*t_1t_2 + ... + (t_n-1 + t_n)*t_n-1t_n, if n even.
If n is odd, t_1 t_2...t_n <- 0t_1t_2...t_n and do a(0t_1 t_2...t_n).
Comments